Average Error: 30.8 → 0.1
Time: 12.9s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.8

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied difference-of-sqr-130.8

    \[\leadsto \log \left(x + \sqrt{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}}\right)\]

  4. Applied sqrt-prod0.1

    \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x - 1}}\right)\]

  5. Final simplification0.1

    \[\leadsto \log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Details

Time bar (total: 12.5s)Debug log

sample178.0ms

Algorithm
intervals

simplify114.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

114.0ms
(log (+ x (sqrt (- (* x x) 1))))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 31.2b

localize13.0ms

Local error

Found 2 expressions with local error:

30.1b
(sqrt (- (* x x) 1))
0.0b
(+ x (sqrt (- (* x x) 1)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

2.0ms
(sqrt (- (* x x) 1))
1.0ms
(+ x (sqrt (- (* x x) 1)))

series75.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

47.0ms
(+ x (sqrt (- (* x x) 1)))
29.0ms
(sqrt (- (* x x) 1))

simplify1.2s

Counts
13 → 31
Calls

13 calls. Slowest were:

333.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
146.0ms
(- (* 2 x) (+ (* 1/8 (/ 1 (pow x 3))) (* 1/2 (/ 1 x))))
111.0ms
(sqrt (sqrt (- (* x x) 1)))

prune260.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize30.0ms

Local error

Found 2 expressions with local error:

0.5b
(* (sqrt (+ x 1)) (sqrt (- x 1)))
0.0b
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 43
Calls

2 calls. Slowest were:

11.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
3.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

series81.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

44.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
37.0ms
(+ x (* (sqrt (+ x 1)) (sqrt (- x 1))))

simplify1.8s

Counts
34 → 49
Calls

34 calls. Slowest were:

615.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))))
393.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
140.0ms
(- (* 2 x) (+ (* 1/8 (/ 1 (pow x 3))) (* 1/2 (/ 1 x))))

prune522.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 3 expressions with local error:

5.6b
(exp (log (* (sqrt (+ x 1)) (sqrt (- x 1)))))
0.5b
(* (sqrt (+ x 1)) (sqrt (- x 1)))
0.0b
(+ x (exp (log (* (sqrt (+ x 1)) (sqrt (- x 1))))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
3 → 66
Calls

3 calls. Slowest were:

10.0ms
(exp (log (* (sqrt (+ x 1)) (sqrt (- x 1)))))
5.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
2.0ms
(+ x (exp (log (* (sqrt (+ x 1)) (sqrt (- x 1))))))

series131.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

51.0ms
(exp (log (* (sqrt (+ x 1)) (sqrt (- x 1)))))
41.0ms
(* (sqrt (+ x 1)) (sqrt (- x 1)))
39.0ms
(+ x (exp (log (* (sqrt (+ x 1)) (sqrt (- x 1))))))

simplify2.7s

Counts
53 → 75
Calls

53 calls. Slowest were:

626.0ms
(* (* (* (sqrt (+ x 1)) (sqrt (+ x 1))) (sqrt (+ x 1))) (* (* (sqrt (- x 1)) (sqrt (- x 1))) (sqrt (- x 1))))
407.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))
354.0ms
(- (+ (* 1/2 (/ (pow x 2) (sqrt -1))) (sqrt -1)) (* 1/8 (/ (pow x 4) (pow (sqrt -1) 3))))

prune889.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes39.0ms

Accuracy

0% (0.0b remaining)

Error of 0.1b against oracle of 0.1b and baseline of 0.1b

bsearch2.0ms

end0.0ms

sample4.4s

Algorithm
intervals